On generalized covering and avoidance properties of finite groups and saturated fusion systems

A subgroup A of a finite group G is said to be a CAP-subgroup of G, if for any chief factor $H/K$ of G, either $AH=AK$ or $A\cap H=A\cap K$. Let p be a prime, S be a p-group and $F$ be a saturated fusion system over S. Then $F$ is said to be supersolvable, if there exists a series of S, namely $1=S_0⩽S_1⩽\cdots ⩽S_n=S$, such that $S_{i+1}/S_i$ is cyclic, and $S_i$ is strongly $F$-closed for any $i=0,1,\cdots ,n$. In this paper, we first introduce the concept of strong p-CAP-subgroups, and investigate the structure of finite groups under the assumptions that some subgroups of G are partial CAP-subgroups or strong (p)-CAP-subgroups of G, and obtain some criteria for a group G to be p-supersolvable. After that, we investigate the characterizations for supersolvability of $F_S\left(G\right)$ under the assumptions that some subgroups of G are partial CAP-subgroups or strong (p)-CAP-subgroups of G, and obtain some criteria for a fusion system $F_S\left(G\right)$ to be supersolvable. The above results improve some known results and develop some new results about CAP-subgroups from fusion systems.